Image processors (scanners, copiers, fax machines, etc) convert a visible image on a document or photograph, or an image in a transparent medium, into an electronic form suitable for copying, storing or processing by a computer. An image processor may be a separate device or may be a part of a copier, part of a facsimile machine, or part of a multipurpose device. In general, image processors use an optical lens system or optical waveguide system to focus an image onto an array of photo sensors. In some processors, a Gradient Index Lens Array (GILA) is employed. A GILA is a row of lenses that goes across the width of the page being processed. One of the issues with the utilization of a GILA is that the farther out of focus the image moves the more the image has a tendency to be seen by multiple lenses. This causes the creation of aberrations known as “echoes” or the “echo effect”. Because the array of lenses is in a row, the echo effect is one-dimensional in nature, occurring only in the horizontal direction. As a result of the echo effect, a “de-echoing” process is desirable in order to minimize the presence of these aberrations or echoes.
De-echoing belongs to the class of de-convolution tasks. In this case the convolution that has to be undone is assumed to have a train of pulses as a kernel. These pulses usually have varying intensities and they occur a fixed distance d from neighboring pulses. Since the out-of-focus distance is known, the convolution kernel is known.
For a better understanding, please refer to FIG. 1A-FIG. 1D. In each Figure, the vertical axis represents an intensity value y and the horizontal axis represents a number of pixels d. FIG. 1A shows a 1 dimensional input signal (image) 100. FIG. 1B shows a train of pulses 110 with variable amplitudes. In this particular example the distance between pulses is d=6. FIG. 1C shows a train of echoes 120 which are the result of convolving the input signal 100 with the train of pulses 110.
It should also be noted that when an input signal is convolved with a train of pulses where d=1, the resulting signal is not referred to as a train of echoes but rather a blurred version of the input signal (image). Likewise, the “train of pulses” is not referred to as such, but rather as a blurring kernel. FIG. 1D shows a resulting signal 130 when the input signal is convolved with a train of pulses with d=1. In either case (de-echoing or de-blurring), the convolution kernel is known and de-convolution is needed in order to minimize the presence of these aberrations.
A conventional solution for the de-convolution of a known kernel is to divide the Fourier coefficients of the measured signal by the Fourier coefficient of the kernel and apply an inverse Fourier transform. However, this solution is unstable in the presence of noise. Other techniques involve the implementation of iterative algorithms that require substantial computation complexity.
In general, what is needed is a method and system that addresses the above-referenced problems associated with the de-convolution of an input image. Additionally, the method and system should be simple, inexpensive and capable of being easily adapted to existing technology. The present invention addresses these needs.